A Space-Time Orbifold: a Toy Model for a Cosmological Singularity

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A Space-Time Orbifold: A Toy Model for a Cosmological Singularity

1 2 Vijay Balasubramanian, ∗ S. F. Hassan, † 2 1 Esko Keski-Vakkuri, ‡ and Asad Naqvi §

1David Rittenhouse Laboratories, University of Pennsylvania Philadelphia, PA 19104, U.S.A.

2Helsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland

Abstract We explore bosonic strings and Type II superstrings in the simplest time dependent backgrounds, namely orbifolds of Minkowski space by time reversal and some spatial reﬂections. We show that there are no negative norm physical excitations. However, the contributions of negative norm virtual states to quantum loops do not cancel, showing that a ghost-free gauge cannot be chosen. The spectrum includes a twisted sector, with strings conﬁned to a “conical” singularity which is localized in time. Since these localized strings are not visible to asymptotic observers, interesting issues arise regarding unitarity of the S- matrix for scattering of propagating states. The partition function of our model is modular invariant, and for the superstring, the zero momentum dilaton tadpole vanishes. Many of the issues we study will be generic to time-dependent cosmological backgrounds with singularities localized in time, and we derive some general lessons about quantizing strings on such spaces.

1 Introduction

Time-dependent space-times are difficult to study, both classically and quantum me- chanically. For example, non-static solutions are harder to find in General Relativity, while the notion of a particle is difficult to define clearly in field theory on time- dependent backgrounds. Quantum mechanical strings propagating on time-dependent

∗[email protected] †fawad.hassan@helsinki.ﬁ ‡[email protected].ﬁ §[email protected]

1 spaces can develop many subtle problems including difficulties with unitarity and ghosts in the physical spectrum. Nevertheless, the apparent observation of a cosmological constant from supernovae measurements [1], and an attendant expansion of the universe, requires us to understand clearly how time dependence of cosmological backgrounds is incorporated into string theory. In related theoretical developments, recent work has explored the physics of de Sitter space [2], as well as new pictures of the early universe in which a collision of branes forms the observable cosmic structures [3]. In the latter models, and in the pre-big bang scenarios [4], a stringy resolution of an initial singularity is proposed to permit an extension of space-time to an era before the big bang. In view of all this it is worthwhile to investigate perturbative string theory in singular cosmological backgrounds. Perturbative string theory is most easily studied in flat, translationally invariant space. The simplest non-homogeneous spaces in which it is well-defined are orbifolds of flat space in which some Euclidean directions are quotiented by a discrete subgroup of the isometry group [5]. When the action of the discrete group has fixed points, the orbifold has conical singularities, as well as new light states (the so-called twisted sectors) which are confined to these defects. Condensing twisted sector states can 4 resolve the conical singularities in many cases such as the classic example R /Z2 where four Euclidean directions are identified under reflections. Can we find consistent backgrounds in string theory by identifying points in space- time rather than just in space? One simple example is the BTZ black hole of three 1 dimensional gravity which is obtained by quotienting AdS3 by a boost [6]. Such orbifolds bear a relation to the kinds of identifications discussed in the context of resolving singularities separating contracting and expanding phases of some cosmological models [3]. Likewise, some coset WZW models are consistent time-dependent string backgrounds [9]. Also, string theory on orbifolds with time identified under t t + 1 (i.e., circular time) has been studied in [10] and the resulting time-like → T-duality has been studied in [11]. Space-time singularities in string theory were studied in [12]. In this paper, we will seek simple models of time-dependent spaces and of cosmological singularities by constructing space-time orbifolds in which we identify space-time under both time reversal and reflections in some directions. Gen- erally speaking, string theories defined on such spaces are threatened by a number of pathologies including potential ghosts in the physical spectrum and problems with unitarity. In fact, all known proofs of the no-ghost theorem explicitly require time- independent backgrounds [13]. Also, supersymmetry is generally broken and so there may be a danger of tadpoles at one loop and instabilities like tachyons could occur. Part of our goal is to explore the many subtleties that beset such constructions in string theory.

1 The consistency of string theory on AdS3 itself is nontrivial, e.g. for the no-ghost theorem and modular invariance see [7]. For work on string theory in BTZ black holes, see e.g. [8].

2 1,d We study bosonic and Type II superstrings on R /Z2, in which we have identified space-time by time reversal and reflections. When d = 0, only time is identified and the space has an initial singularity at t = 0. When d 1 the background geometry ≥ is a space-time cone with a “conical” singularity at t = x = = xd = 0. String 1 ··· theory on such spaces is defined by projecting onto the sector of the Hilbert space that is invariant under these discrete transformations, and including possible twisted

sectors localized at the orbifold fixed point at t = x = = xd =0,andwhich 1 ··· therefore do not propagate. After this projection, quantum mechanics is consistent with closed time-like loops in the geometry. We find that the physical states are ghost-free when d +1 9 for the bosonic string and that there is no restriction on d ≥ for the superstrings. In Type II superstrings, when d + 1 = 4, there is a “massless” twisted sector in which physical states satisfy the on-shell condition p~ 2 =0.2 It is | | possible that condensing these states would resolve the conical singularity, and so we focus on the d +1=4case. We compute the partition function when d + 1 = 4 and find that it is zero. Likewise the one loop zero-momentum tadpoles vanish suggesting that we have a consistent string background at this order in string perturbation theory. However, negative norm states (although not present in the on-shell physical spectrum) make a contribution to the partition function – their virtual effects in the twisted sector do 4 not cancel between the matter and ghost sectors as they do in the standard R /Z2 orbifold. This shows that it is not possible to choose a ghost-free gauge in which all computations are carried out in terms of positive norm states. We expect that this will be generally true for string theory in time-dependent backgrounds.3 We conclude the paper by discussing several novel subtleties introduced by the localization in time of a sector of physical states, and by summarizing lessons learned from our work about time-dependent backgrounds and cosmological singularities in string theory.

2 Space-time orbifolds

We study space-time orbifolds constructed by identifying Minkowski space under time reversal and reflections in some spatial directions. As we will see below, the resulting geometry can be interpreted as a space-time cone. After the identifications the covering space has some closed time-like loops. However, because the orbifold prescription projects onto states in Hilbert space that are symmetric under the identifications, quantum mechanical evolution remains consistent. 2Actually, this implies that p~ = 0 since the twisted sector states are localized in time and so only carry momenta in the un-orbifolded Euclidean directions. 3This is reminiscent of mixing between the matter and ghost sector CFTs discussed in [14].

3 t t I II

III IV x

(a) (b)

Figure 1: A space-time cone.

2.1 Conical space-time geometries Consider identifying time (X0)andd spatial directions under reﬂections:

Xa Xa (a =0, ,d) , (1) →− ··· leaving all other directions unaffected. Fig. 1 shows the resulting space-time cone when d = 1. Points in opposite quadrants of the X T plane are identified as in − Fig. 1. Therefore the quadrants II and IV (or I and II) may be taken as “fundamental” regions with independent physics. Identifying these regions along the T (or X) axis produces the cone in Fig. 1b with a singular point at T = X =0. The proper distance on the covering space between a point (T,X) and its image ( T, X)isds2 =4(X2 T 2). This is time-like in the region inside the light cone − − − emanating from the point T = X = 0 on the covering space. As a result there are closed time-like curves in this geometry, such as the one in Fig. 1a. In the orbifold construction which we will describe below, such loops do not pose a fundamental problem since we are instructed to project to states in the Hilbert space that are invariant under the space-time identifications, i.e., we project onto quantum mechanical wave-functions that satisfy ψ(x, t)=ψ( x, t). A picture of time evolution on the − − cone is provided in Fig. 2a where we have folded regions II and IV along the X-axis and identified the negative and positive directions along the time axis, to make a cone. It is natural then to describe the evolution of states on the cone with respect to the time direction inherited from the positive time direction in quadrants II and IV of the parent manifold. The line x = 0 appears to have time “running both ways”, but this is simply because we have projected onto states that are time reversal invariant on the X =0axis. Constructing the cone by gluing the X axis of quadrants I and II yields a similar picture with two “sheets” glued together on the T and X axis. At first sight the time inherited from the covering space gives evolution moving “up” on both sheets

4 t t

x x (a) (b)

Figure 2: Time evolution on the cone.

in Fig 2b, with the boundary condition that the wave-functions on both sheets ap- proach the same value on a big-bang-like surface at T = 0. However, on the X axis

of the covering space the orbifold identiﬁcations also imply that ∂ψ(x, t)/∂t t = | =0 ∂ψ( x, t)/∂t t . Therefore, on the cone, with time evolving “up” on both sheets, − − | =0 although wave-functions on both sheets agree on the initial surface, their time deriva- tives are opposites of each other. Therefore it seems more natural once again to describe the evolution of states with respect to a continuous time as in Fig. 2a.

2.2 Euclidean world-sheets and Lorentzian backgrounds As we have discussed, we will construct string theory on our space-time orbifold by projecting onto states of strings in Minkowski space that are invariant under the discrete identifications. In Lorentzian space-times the signature of the string worldsheet must be ( 1, 1) in order for classical string propagation to exist. (The 2d − equations of motion are solved by equating the worldsheet metric with the metric induced from space-time.) Nevertheless, the standard techniques of string theory involve analytically continuing the worldsheet to Euclidean signature in order to exploit the techniques of two-dimensional conformal field theory and complex geometry. In static backgrounds we might imagine continuing the space-time to Euclidean signature at the same time, but this is not possible in time-dependent backgrounds such as ours. In this paper we will take the attitude that perturbative string theory is a2d worldsheet CFT and that this field theory is defined by analytic continuation from Euclidean signature. In performing this continuation of the 2d field theory it is not necessary to continue the worldsheet fields which are the coordinates of the string in space-time. Hence, although our background is time-dependent, we will freely continue the worldsheet to Euclidean signature.

5 3 Bosonic string theory on the Lorentzian orbifold

Before studying superstrings on space-time orbifolds we examine the 26-dimensional 1,d bosonic string propagating on R /Z2. This already contains the distinctive features of the Lorentzian orbifold. In particular, we show that it is possible to obtain a ghost- free physical spectrum and a modular invariant partition function, but that virtual negative norm states make un-cancelled contributions to quantum loops. This is a reﬂection of the time dependence of the string theory background.

Consider flat 26-dimensional Minkowski space with points identified under the Z2 action, Xa Xa (a =0 d); Xi Xi (i = d +1 25). (2) →− ··· → ··· This action has a fixed (25 d)-dimensional hyper-plane, given by Xa =0.Toget − consistent string propagation on this space-time, we project the conventional bosonic string Hilbert space onto its Z2 invariant subspace. This gives the untwisted sector of the orbifold theory. In addition, there is a twisted sector corresponding to strings that are closed only under the identifications made by the orbifold group. Again, we project out twisted sector states that are not invariant under the orbifold action. The twisted strings are trapped around the tip of the cone in Fig. 1b, which is a (25 d)- − dimensional hyper-plane localized at an instant in time. The untwisted strings can propagate in the bulk. The orbifold above has the novel feature that it includes a reflection in the time direction, destroying the global time-like isometry of flat space-time. This means that we cannot perform quantization by going to light-cone gauge. The alternative is to use the covariant BRST formalism. However, in the absence of a light-cone gauge choice, the absence of negative-norm states in the physical spectrum is no longer evident, especially in view of the non-applicability of the known proofs of no-ghost theorem [13]. In the following, we will mostly be concerned with this issue. In the covariant formalism, we work with world-sheet fields Xµ (µ =0, , 25) and the ··· reparameterization ghosts b and c. In the untwisted sector Xµ(σ +2π, τ)=Xµ(σ, τ ), and the mode expansion is4 µ µ µ µ µ αn in(τ σ) αñ in(τ+σ) X = x + p τ + i e− − + i e− . (3) n=0 n n=0 n X6 X6 The (tachyonic) ground state pa,pi carries momentum in both orbifolded and un- | i orbifolded directions and the Hilbert space of states is constructed by acting with creation operators on the ground state. Half of the states with non-zero pa are projected a i a i a i a i out of the spectrum. For example, of the states α 1α˜ 1 p ,p and α 1α˜ 1 p ,p , a i a i − a −i | i − − |− a i only the linear combination α 1α˜ 1( p ,p p ,p ) is retained. When p =0, − − | i−|− i only the Z2 invariant combinations of the oscillators acting on the vacuum are kept. a i i a b i Hence, α 1α˜ 1 0,p is projected out but α 1α˜ 1 0,p is retained. − − | i − − | i 4 We will work in α0 = 2 units in this paper.

6 In the twisted sector, the fields Xa satisfy the anti-periodic boundary condition Xa(σ +2π)= Xa(σ), with the mode expansion given by, − a a a αr ir(τ σ) α˜r ir(τ+σ) X = i e− − + i e− , (4) r r r r X X where r is half odd-integral. The twisted sector is localized at the orbifold fixed plane at Xa = 0. In particular, for any value of the parameter τ, the twisted string worldsheet does not propagate too far out in time X0. This is similar to an instanton. The mode expansion for Xi in directions transverse to the orbifold is the same as in the untwisted sector. Consequently, the ground state in the twisted sector carries a momentum pi only in the transverse directions. The Hilbert space is built by acting with creation operators on the ground state and projecting onto the Z2 invariant subspace. The ghosts b and c are not affected by the orbifold and have the same mode expansions in both sectors:

in(τ σ) in(τ σ) b(σ, τ )= bne− − ,c(σ, τ )= cne− − ;(5) n n X X and similarly for right-movers ˜b andc ˜.

3.1 Physical states

The BRST operator QB is given by

m ˜m (m n) ˜ QB = (cnL n+˜cnL n)+ − :(cmcnb m n+˜cmc˜nb m n):+a (c0+˜c0), (6) n − − m,n 2 − − − − X X where Lm are the Virasoro generators in the matter sector and a is the zero point energy. Physical states are elements of the BRST cohomology, i.e.,theyobeyQB ψ = | i 0 subject to the equivalence relation ψ ψ + QB φ ,where φ is an arbitrary | i∼| i | i | i state. It is now easy to see that the physical spectrum does not contain negative norm states. In the untwisted sector, after the orbifold projection, the states form a subspace of the Fock space of the parent theory. Furthermore , the orbifold action (2) commutes with the BRST operator (6). This means that the space of physical states of the orbifold theory is a subspace of the space of physical states of the parent theory, and hence is free of negative norm states. More explicitly, for pa = 0, one can easily 6 establish a correspondence between states in the parent theory and those of the orbifold theory by appropriately choosing symmetrized or antisymmetrized momentum wave-functions. To see that the twisted sector physical states do not have negative norms, recall that the BRST condition QB ψ =0,alongwithb ψ = 0, implies (see, for example, | i 0| i [15]) (Lm + Lgh a) ψ =0, (7) 0 0 − | i 7 gh where L0 is the ghost Virasoro generator. In terms of the twisted sector number operators, we have

25 d m gh 1 i ∞ ∞ L0 + L0 = p pi + n(Nbn + Ncn + Nin)+ rNar , (8) 2 n a X=1 i=Xd+1 r=1X/2 X=0 26 (d+1) 2 d+1 15 d and the twisted sector zero-point energy is a = − − = − . Then for a 24 − 48 16 physical state ψ,pi ,withmomentumpi in the un-orbifolded directions, this implies | i 25 d 1 i ∞ ∞ 15 d p pi + n(Nbn + Ncn + Nin)+ rNar = − . (9) 2 n a 16 X=1 i=Xd+1 r=1X/2 X=0 Since the left hand side is always positive, d is restricted to d 15 in order to allow ≤ for any physical states in the twisted sector. Furthermore, since (15 d)/16 < 1, a − twisted sector physical state will not contain c, b and Xi excitations. For 1 d 7, ≤ ≤ the physical spectrum will always contain a negative norm state corresponding to 0 α 1/2. However, for d 8 there are no negative norm states in the twisted sector − ≥ physical spectrum which, for 15 d 8, contains only the ground state 0,pi .In ≥ ≥ | i particular, pi = 0 for d = 15. 5

3.2 Partition function and virtual ghosts Although there are no negative norm physical states (for the right range of d), the orbifold theory may still contain negative norm virtual states running in loops. This can be studied by looking at the one-loop partition function. Before considering the orbifold case, we recall the partition function of the closed bosonic string in 26- dimensional Minkowski space,6

H H V26 1 26 1 m 4 Z(τ,τ¯)=Trq L q¯ R q 24 (1 q ) 12 1 m 2 ∼ τ2 q 24 (1 q ) ×| m − | | m − | Y V 1 24 = 26 Q , 12 1 m 2 τ2 q 24 (1 q ) | m − | 5For d = 15, the state in the twisted sector is physical only when pi = 0. If this state at pi =0 Q 6 were BRST exact, it would be orthogonal to all other physical states. Amplitudes involving such (9 d) i a state would then have to be proportional to δ − (p ). As argued in [15], since amplitudes in ﬁeld theory and string theory never have this kind of a behaviour, such a state with pi = 0 should not be part of the physical spectrum. However, this is not true for the twisted sector state on the Lorentzian orbifold. This is because the state with non-zero pi is not BRST exact since it is not even BRST closed (Q pi =0 = 0), as it does not satisfy the Virasoro constraint. So the above B| 6 i6 argument does not apply and the zero-momentum physical twisted state should be retained. 6More precisely, the deﬁnition of Z(τ,τ¯)is

Z(τ,τ¯)=Tr( 1)F c b c˜ ˜b qHL q¯HR (10) − 0 0 0 0 where ( 1)F anticommutes with all the ghost ﬁelds. In the following, we implicitly assume that the − trace is taken with ( 1)F c b c˜ ˜b inserted. − 0 0 0 0 8 ˜ 2πiτ where HL = L a, HR = L a, a is the zero point energy, and q = e . V is a 0 − 0 − 26 space-time volume factor related to the continuum normalization of the momentum integral and m is a short hand for m∞=1. Here, the contributions from the negative and positiveQ norm ghost states cancelQ the contributions from the time-like and one space-like oscillators, respectively, giving the same result as we would get in the light- cone gauge. To verify that this really is how the cancellations work, one can compute the following closely related quantity,

1 m 2 1 m 2 V q 24 (1 q ) q 24 (1 + q ) s HL HR 26 m m S(τ,τ¯)=Tr( 1) q q¯ 12 | − | 25| | τ 1 m 1 m − ∼ 2 q 24 Q(1 q ) 2 q 24Q (1 + q ) 2 | m − | | m | V Q 1 24 Q = 26 . 12 1 m 2 τ2 q 24 (1 q ) | m − | The insertion ( 1)s ensures that negative normQ states contribute with a negative sign − in the trace. The equality Z(τ,τ¯)=S(τ,τ¯) then reﬂects the fact that the negative norm ghost contribution really did cancel that of the time-like oscillators.7 1,d We now proceed to the partition function of the orbifold R /Z2. A partition function, being a vacuum amplitude, is a space-time scalar. Therefore, the trace in µ it extends to the space-time index of the states, i.e., the conjugate to α n p appears µ − | i as p αµ,n. The commutators then involve δ rather than ηµν and time-like oscillators h | ν contribute in the same way as space-like ones. The partition function then has the 1+d same form as that of the Euclidean orbifold R /Z2,andisgivenby 1+ˆg 1+ˆg Z(τ,τ¯)=Tr qHL q¯HR +Tr qHL q¯HR (11) U 2 T 2 24 (d+1) − V25 d 1 = −  1 m 2  2 √τ2 q 24 (1 q ) | m − |  VQ  V d+1 + d+1 1 m 2 d+1 1 m 2(d+1) × "(√τ2 q 24 (1 q ) ) q 24 (1 + q ) | m − | | m | 1 1 + Q + Q , (12) 1 m 1 2(d+1) 1 m 1 2(d+1) q− 48 (1 q − 2 ) q− 48 (1 + q − 2 ) # | m − | | m | whereg ˆ is the Z2 action onQ the Hilbert space and TrU Qand TrT denote traces over untwisted and twisted sector states respectively. Note that since twisted sector states do not carry momentum in the d + 1 orbifolded directions, their contributions do not contain the corresponding continuum normalization factor Vd+1 for the momentum (d+1) integral. The partition function is modular invariant with Vd+1 =2− .Itis d+1 instructive to compare this with the torus orbifold T /Z2. In that case, since the momenta in the orbifold directions are discrete, the normalization is trivial. But then the orbifold has 2d+1 twisted sectors associated with as many ﬁxed points which results in modular invariance. 7We thank C. Vafa for this argument.

9 Although the expression for the partition function of the Lorentzian orbifold 1,d d+1 R /Z2 is the same as that of the Euclidean orbifold R /Z2,theyembodyvery different physics. In the Euclidean orbifold, the negative norm ghost contribution always cancels against time-like oscillators, indicating the possibility of choosing a gauge (the light-cone gauge), in which there are no negative norm states. However, 1,d in the Lorentzian orbifold R /Z2, virtual negative norm states make uncancelled contributions to the partition function. As a check of this, one can again look at 1+ˆg 1+ˆg s HL HR s HL HR S(τ,τ¯)=TrU ( 1) q q¯ +TrT ( 1) q q¯ . (13) − 2 − 2 The difference, if any, between S(τ,τ¯)andZ(τ,τ¯) can only arise because of differing contributions from the negative norm states and therefore, we concentrate on these parts of Z(τ,τ¯)andS(τ,τ¯). The contribution from negative norm states in the left moving sector to various terms in (11,13) is given by:

(1 qn) (1 qn) s L0 s L0 Tr U ( 1) q ∓ n , Tr U ( 1) gqˆ ∓ n , ± ∼ n (1 q ) ± ∼ n (1 q ) Y ∓ Y ± (1 qn) (1 qn) s L0 s L0 Tr T ( 1) q ∓ , Tr T ( 1) gqˆ ∓ . (14) n+ 1 n+ 1 ± ∼ n (1 q 2 ) ± ∼ n (1 q 2 ) Y ∓ Y ± The upper signs in the four expressions correspond to contributions of negative norms states to Z(τ,τ¯) and the lower signs correspond to their contributions to S(τ,τ¯). The factor (1 qn) in the numerator is the contribution from the negative norm ghost states ∓ and the factor in the denominator is the contribution from the time-like oscillators.8 From these expressions, it is clear the the contributions of the negative norm states to Z(τ,τ¯)andS(τ,τ¯)arenotthesame.HenceS(τ,τ¯) = Z(τ,τ¯), which explicitly 6 shows that the contributions of virtual negative norm states in the partition function do not cancel on the Lorentzian orbifold. This implies that a ghost-free gauge for string theory in such a background does not exist. This is perhaps not surprising: we cannot choose the light cone gauge because our orbifold involves a reflection in the time direction. One might have thought that there is some other gauge in which all calculations can be done in terms of positive norm states, but the analysis above shows that such a gauge does not exist. Nevertheless, as we have shown, there are no negative norm physical states on the orbifold. We expect that these features are generic for theories in time dependent backgrounds. Since b and c are reparameterization ghosts, their periodicities on the world-sheet torus are fixed by the theory. However, suppose that we regard the Xµ as describing simply a free field theory on the orbifold. Then, in principle, we can introduce a (b, c) system, not as reparameterization ghosts, but with the sole purpose of removing the

8 1 Note that for the Euclidean orbifold, the contributions from the time-like oscillator is 1 qn which cancels the contribution from the negative norm ghost oscillators resulting in the equality∓ Z(τ,τ¯)=S(τ,τ¯) for the Euclidean orbifold.

10 negative norm states. Then, by assigning appropriate periodicities to the ghosts, depending on the Xµ boundary conditions, it is possible to fully cancel the contributions of negative norm states in all sectors of the theory. However, this will not be a string theory.

Summary We have found that there are no negative norm physical states in the bosonic string theory on the Lorentzian orbifold R1,d/Z when d +1 9, and the 2 ≥ partition function is modular invariant. However, negative norm virtual states make uncancelled contributions to quantum loops. This implies that it is not possible to choose a gauge in which all computations are done in terms of positive norm states. For 9 d +1 16, the ground state in the twisted sector pi carrying momentum ≤ ≤ 2 15 d | i in the un-orbifolded directions is physical with p~ = − .Ford +1> 16, there are | | 8 no physical states in the twisted sector.

4 Type II superstrings on the Lorentzian orbifold

We will next move on to type II superstrings. Because the orbifold involves time, we will work in the covariant RNS formulation. Now the orbifold action is Xa Xa ,Xi Xi ,ψa ψa ,ψi ψi , (15) →− → →− → where, a =0, ,dand i = d +1, , 9. For technical reason, we will always consider ··· ··· d odd. We ﬁrst look at the untwisted sector. Here, the fermions have the standard µ µ ir(τ σ) mode expansions: ψ (σ )= r ψr e− − , with similar expressions for left-movers ˜µ − 1 ψ (σ+). The sum is over r PZ + 2 in the NS sector and r Z in the R sector. The ∈ ∈ 1 bosons have the mode expansions (3). The zero point energy a = aB + aF is a = 2 in the NS sector and a = 0 in the R sector. The NS sector ground state is a tachy- a i a i onic scalar p ,p NS, whereas the R ground state is a massless spinor p ,p R.The | i | i orbifold operation acts on the R vacuum as

a i d+1 0 1 d a i p ,p R (Γ ) Γ Γ Γ p ,p R. (16) | i → 11 ··· |− i After the orbifold projection, the invariant states have momentum wave-functions of definite symmetry, pa,pi pa,pi , depending on the (d + 1)-dimensional chirality | i±|− i of the R ground state and the oscillator numbers. As in the bosonic orbifold of the previous section, the physical untwisted orbifold states form a subspace in the space of physical states of the parent type II theory. Consequently, the untwisted sector is free of physical negative norm states. The supersymmetry of the physical untwisted spectrum (for odd d) can be illus- trated as follows. Let (1,9) denote an SO(1, 9) spin-field of definite chirality that re- S (1,9) lates the NS and R ground states in type II theory, p R p NS. Then the space- | i ∼S | i φ/2 (1,9) time supersymmetry current in the unorbifolded theory has the form J e− , ∼ S 11 φ/2 (1,9) where e− is the spin-field for the β,γ ghost system. Suppose that has positive n S chirality and we denote SO(n) spinors of 1 chirality by ( ). As the orbifold breaks ± S± SO(1, 9) to SO(1,d) SO(9 d), the positive chirality spin-field decomposes as × − (1,9) (1,d) (9 d) (1,d) (9 d) + = + + − + − . S S ⊗S S− ⊗S− Then, under the orbifold projection (16), the piece with positive SO(1,d) chirality φ/2 (1,d) (9 d) survives and the orbifold inherits a supercurrent Jorb e− − from the ∼ S+ ⊗S+ parent theory. This proves the supersymmetry of the untwisted sector, while showing that the amount of supersymmetry has been reduced. We now turn to the twisted sector. The twisted bosons have the mode expansion (4). Fermions in the twisted sector satisfy boundary conditions: NS : ψa(σ +2π)=ψa(σ) ,ψi(σ +2π)= ψi(σ) ; (17) − R: ψa(σ +2π)= ψa(σ) ,ψi(σ +2π)=ψi(σ) . (18) − These lead to the mode expansions:

a a in(τ σ) i i ir(τ σ) NS : ψ (σ )= ψn e− − ,ψ(σ )= ψr e− − ; (19) − − n Z r Z+ 1 X∈ ∈X 2 a a ir(τ σ) i i in(τ σ) R: ψ (σ )= ψr e− − ,ψ(σ )= ψn e− − . (20) − − r Z+ 1 n Z ∈X 2 X∈ The periodicities and mode expansions are reversed along the orbifolded directions compared to the unorbifolded ones. The twisted NS sector has fermion zero modes along the orbifold and the corresponding ground state pi T is a SO(1,d) spinor and | iNS a SO(9 d) scalar. The twisted R sector ground state pi T is a spinor under SO(9 d) − | iR − and a scalar under SO(1,d). Some more details can be found in the Appendix B.

Using the mode expansions, the Virasoro generators Lm and the worldsheet su- percurrents Gr and Fn can be worked out. These are summarized in Appendix A. To identify the physical spectrum, one also needs the zero point energies, a = aB + aF . In the NS sector, the worldsheet bosonic and fermionic sectors contribute as, d +1 9 d 2 d +1 9 d 2 aB = + − ,aF = + − . (21) − 48 24 − 24 − 24 48 − 48 Here, 2/24 is the contribution from the b, c ghosts and 2/48 is the contribution − − from the NS sector β,γ ghosts. In the twisted Ramond sector, aB is as above and the fermions give, (9 d) d +1 2 aF = − + + , (22) − 24 48 24 where 2/24 is from the Ramond sector β,γ ghosts. In total then, 3 d a = aB + aF = − (Twisted , NS) , (23) 8 a = aB + aF = 0 (Twisted , R) . (24) The zero point energy vanishes for any value of d in the twisted Ramond sector.

12 4.1 Twisted sector physical states The content of the twisted sector physical spectrum is determined by the super- Virasoro constraints,

(Lm aδm) phys =0 (m 0), − | i ≥ (25) 1 Gr phys =0 (r , NS) ,Fn phys =0 (n 0 , R) , | i ≥ 2 | i ≥

with the generators given in Appendix A. As in the bosonic case, the L0 constraint gives

i 3 d p pi + lNl = − (NS sector), (26) 8 Xl = 0 (R sector) . (27)

Here pi is the momentum carried by the twisted sector state in the unorbifolded

direction, and l lNl schematically represents the combined sum over the bosonic, fermionic and ghostP number operators in the twisted sector. Note that the minimum 1 1 non-zero value of this sum is 2 , while the right hand side is always less than 2 .There- fore, physical twisted states cannot have any oscillator excitations. In particular, they will be free of negative norms. In the twisted NS sector, there are no physical states i T i 3 d for d>3. For d 3, the twisted NS ground state p is physical with p pi = − . ≤ | iNS 8 In particular, for the case of d = 3, this ground state has pi =0. This state also trivially satisfies all the other physical state constraints in (25). In the R sector, the only physical state, for any d, is the Ramond ground state at i T zero momentum, p =0R. This also satisfies the remaining constraints in (25). In | i i i T particular, the F constraint gives piΓ p = 0, which is normally the Dirac equation 0 | iR reducing the number of spinor components by half. In our case, since pi =0,itdoes not impose a constraint. Thus, e.g. in d = 3 the twisted R sector vacuum has twice as many components as the twisted sector NS vacuum (see Appendix B). The GSO projection results in the NS sector ground state, pi T ,havingthesame | iNS SO(1,d) chirality in the left and right moving sectors.9 In the twisted R sector, the ground state pi T , has the same (opposite) SO(9 d) chirality in the left and right | iR − moving sector for Type IIB (Type IIA) string theory. In general, the bosonic and fermionic degrees of freedom in the twisted sector will not match. For the special case of d = 3, the twisted sector NS ground state is a chiral spinor of SO(1, 3) and the R sector ground state is a chiral spinor of SO(6). These spinors have different dimensionalities and as a result bose-fermi degeneracy of the space-time spectrum is broken in the twisted sector. 10 9Recall we are consider odd d so chirality is well defined. 10 4 In the case of the Euclidean orbifold R /Z2, the Dirac equation in the Ramond sector reduced the fermionic components by half resulting in a supersymmetric spectrum.

13 4.2 Partition function and tadpoles The one-loop partition function, as in the bosonic case, does not distinguish between space-like and time-like oscillators. Therefore, the result for superstrings on 1,d the Lorentzian orbifold R /Z2 will be the same as that for the Euclidean orbifold d+1 R /Z2. This is in spite of the fact that the spectra in the two cases are very different, especially in the twisted sector. For definiteness, we look at the case of d =3. The torus partition function for the orbifold is given by (1 +g ˆ) (1 +g ˆ) Z(τ,τ¯)=Tr qHL q¯HR +Tr qHL q¯HR , U 2 T 2 whereg ˆ is the representation of the Z2 orbifold action on the Fock space, and TrU and TrT represent traces taken over the untwisted and the twisted sectors. We also need to sum over the four different spin structures of the torus in both sectors. The contributions from the b, c and β,γ ghosts will cancel the contributions from two unorbifolded Euclidean directions. Then, for the d = 3 case, the result after the relative sign factors for the contributions from different spin structures have been chosen, is

2 a a + h a h (h,g) θ θ θ − V 1 (V )1 hZ 1 " b # " b + g # " b g # Z = 6 4 − b ( 1)(a+b+ab) − 2 4 4 (h 1)g 4 2τ2 η η¯ (16) − × − 2η h,gX=0 a,bX=0 a¯ a¯ + h a¯ h θ¯2 θ¯ θ¯ − 1 " ¯b # " ¯b + g # " ¯b g # ( 1)(¯a+¯b+λa¯¯b) − , (28) 4 × ¯ − 2¯η a,¯Xb=0 where λ =0, 1 for type IIA, IIB superstring. This is the same as the Euclidean case (see, for example, [16]). The θ-functions are deﬁned as

a a 1 (n a )2 πib(n a ) θ θ (0 τ)= q 2 − 2 e− − 2 , " b # ≡ " b # | n Z X∈ and Zb is the contribution from the bosonic sector, 1 η2η¯2 Z(0,0) = ,Z(h,g) = , (h, g) =(0, 0). b 2 4 4 b τ2 η η¯ 1 h 1 h 6 θ2 − θ¯2 − " 1 g # " 1 g # − − V6 and V4 are volume factors entering the continuum normalization of the momentum integrals parallel and transverse to the orbifold. h = 0 for the twisted sector and g =0 for terms without the operatorg ˆ. The contributions from the worldsheet fermions vanish in each one of the four (h, g) sectors separately, due to the Jacobi identity,

1 4 4 1 a + hi 1 hi ( 1)a+b+ab θ = θ − , 2 − i " b + gi # − i " 1 gi # a,bX=0 Y=1 Y=1 − 14 1 coupled with θ = 0. Hence Z(τ,τ¯) = 0, without having to fix the relative " 1 # factor V4. The vanishing of the partition function in particular implies its modular invariance. All this looks rather surprising considering the difference between the Euclidean and Lorentzian orbifolds. As in the bosonic case, the difference can be made manifest by inserting, in the partition function, an operator ( 1)s that changes − the sign of all negative norm states. Once again one finds that although the physical spectrum is free of negative norm states, non-physical negative norm states do not decouple in the loops. The vanishing of the partition function implies that there is no dilaton tadpole, at zero energy-momentum [17]. In the absence of the orbifold, a non-zero momentum tadpole vanishes simply by momentum conservation. However, in the space-time orbifold, because of energy-momentum non-conservation at the “conical” singularity, kinematics can allow inserting, on the torus, a dilaton vertex operator carrying non- zero energy and momentum. The vanishing of such tadpoles is not obvious and requires further investigation.

Summary For Type II superstring, we have found that there are no negative norm 1,d physical states on the Lorentzian orbifold R /Z2. The ground state in the twisted NS sector transforms as a spinor of SO(1,d)andascalarofSO(9 d). It is only − physical when d 3 and the momentum it carries in the un-orbifolded directions has 2 ≤3 d to satisfy p~ = − . In the twisted Ramond sector, the ground state is a SO(1,d) | | 8 scalar and SO(9 d) spinor. It is physical for any value of d and its momentum in − the un-orbifolded directions has to vanish: pi = 0. The partition function is modular invariant and the zero momentum dilaton tadpole vanishes at one loop.

5 Discussion

In this article we studied two basic issues in string theory about which very little is known – time-dependent backgrounds and cosmological singularities. We chose the simplest possible spaces exhibiting these phenomena, space-time orbifolds of Minkow- ski space, and showed how simple quotients by time reversal and spatial reflections evade some of the obvious potential pitfalls (tachyons and ghosts in the physical spectrum, zero-momentum tadpoles at one loop, lack of modular invariance etc.). Although there are closed time-like loops in the construction, quantum mechanical evolution is consistent because the orbifold prescription projects onto states that are invariant under the discrete identification. How is an S-matrix defined when a class of physical states is localized in time? An asymptotic observer in models such as ours only observes transition amplitudes between the propagating untwisted sector states. Any such amplitude could involve

15 the emission of arbitrarily many twisted sector states which cannot be observed at late or early times. Therefore it appears that the rules for computing transition amplitudes in space-time orbifolds will require tracing over emissions of states that are localized in time. If so, pure states scattering off a space-time orbifold singularity could emerge as mixed states due to entanglement with an unobservable twisted sector. Perhaps such a mechanism is responsible for creating the observed large entropy of the universe in a cosmological context. One might wonder whether a loss of unitarity is also implied by the uncancelled contributions of negative norm virtual states in the partition function of our space- time orbifolds. Certainly, this result implies that it is not possible to choose a ghost- free gauge in which all computations are carried out in terms of positive-norm states. We expect that this will be true in many time-dependent backgrounds of string theory – it is at least clear that the ghost-free light-cone gauge cannot be chosen in time- dependent backgrounds. This is in sharp contrast to usual string theories and field theories in static backgrounds. Nevertheless, it is not clear that a loss of unitarity in transition amplitudes is implied. In particular, since our models do not have any physical negative norm states, cutting the one loop diagram will not give a transition amplitude to a ghostlike state. In absence of a general argument connecting negative norm virtual contributions to the partition function and S-matrix unitarity, we require detailed study of amplitudes for propagating untwisted sector states scattering from the orbifold singularity. There are very interesting subtleties in the computation of correlation functions and transition amplitudes on space-time orbifolds such as ours in which the twisted sectors are localized in time. Because of the localization, we do not expect energy (or momentum in any of the orbifolded directions) to be conserved in interactions between the untwisted and twisted sectors. One important consequence is that (unlike usual spatial orbifolds) kinematics does not forbid a finite momentum tadpole appearing at one loop. We can expect that this issue of finite-momentum tadpoles will persist for time-dependent string backgrounds in general. 1,3 One reason for our focus on the R /Z2 orbifold of the superstring is that this orbifold had “massless” twisted sector states with Euclidean momenta satisfying ~p2 =0. 4 In the classic R /Z2 orbifold the massless twisted sector states (for which Lorentzian p~2 = 0) correspond to geometric blowup modes which can resolve the singularity. Some condensates of the twisted sector states correspond to parameters of the Eguchi- Hanson Ricci-flat metric on the smooth manifold obtained by replacing the tip of the 4 R /Z2 cone by a sphere. We might hope that some conical space-time singularities can be resolved by similar condensates of “massless” twisted sector states. When the twisted sector states are tachyonic, we might similarly hope that tachyon conden- sation would resolve the orbifold singularity. Unfortunately, much of the geometric technology of deforming singular manifolds into smooth spaces relies on complex geometry and cannot accommodate a manifold with signature (1,d). For example, the

16 Eguchi-Hanson metric [18] has signature (4, 0) and while one can easily obtain a (2, 2) signature Ricci-flat metric by analytic continuation from it, a (1,3) signature Ricci-flat metric cannot be obtained in this way.11 In order to understand cosmological singularities in string theory it is urgent that we develop the mathematics of resolution of singularities of Lorentzian manifolds.12 In string theory, the quantum mechanics of a relativistic string is used to compute transition amplitudes and an S-matrix for the scattering of conventional multi- graviton states. In view of this we have studied the quantum mechanics of strings on space-time orbifolds. Field theories on such spaces raise several new issues. For example, new singularities can potentially arise in correlation functions of operators at space-like separations if the space-time interval between some operators and the orbifold images of others is time-like or null. The rules for defining field theories in such backgrounds remain to be worked out.13 We conclude here by summarizing some perspectives from this work about time- dependent backgrounds and cosmological singularities in string theory:

String theories deﬁned on time dependent backgrounds run the risk of having • ghosts and tachyons in the physical spectrum.

Even when there are no ghosts in the physical spectrum, negative norm states • can make uncancelled contributions to the partition function. In such cases it is not possible to choose a ghost-free gauge like light-cone gauge. This might lead to loss of unitarity, but a more detailed analysis is needed.

The quantum mechanics of strings on space-time orbifolds can be consistently • deﬁned even if there are closed time-like loops by projecting onto states invariant under the orbifold group. It would be interesting to consider space-time orbifolds without closed time-like curves, but we expect the issues raised here to persist (see [19]).

The resulting orbifolds can be tachyon and ghost-free and typically contain a • twisted sector at a ﬁxed plane localized in time.

Scattering from such an asymptotically unobservable twisted sector could cause • transitions from a pure state to a mixed state, generating entropy. 11A simple generalization of the Eguchi-Hanson metric cannot work because the curvature two form for an Eguchi-Hanson space is self dual, but in (1, 3) signature, the self-duality condition has an extra factor of ’i’. 12Perhaps the geometric diﬃculty in resolving these singularities is related to the fact that the “massless” twisted sector states are not exactly moduli ﬁelds in the low energy theory. They are localized in time and are on shell only at zero momentum. 13We are grateful to Nati Seiberg for a discussion of these issues. Also see [19].

17 Since energy need not be conserved in a time-dependent background, kinematics • does not forbid the production of tadpoles with ﬁnite momentum. Hence, the vanishing of these amplitudes must be checked to conﬁrm the existence of a valid solution to string theory.

We expect to return to many of the issues laid out above in a future publication. Acknowledgments

We thank J. Christian, M. Cvetic, E. Gimon, P. Kraus, D. Minic, B. Ovrut, N. Seiberg, G. Shiu, C. Thorn, C. Vafa, and E. Witten for useful conversations. E.K-V. has been in part supported by the Academy of Finland, and thanks the University of Pennsylvania for hospitality. V.B. dedicates this work to Ravi Nicholas who arrived while our paper was being written. V.B. and A.N. were supported by DOE grant DOE-FG02-95ER40893.

A Twisted sector superconformal generators

1,d Here we list the super conformal generators in the twisted sector of the R /Z2 orbifold theory. The Virasoro generators are given by

NS B F,NS R B F,R Lm = Lm + Lm ,Lm = Lm + Lm (29)

where, in the twisted sector,

B 1 a b i i Lm = : α n 1 αm n 1 ηab + α nαm+n : (30) 2 + + 2 2 n Z − − − X∈ and

F,NS 1 m a b 1 m i i Lm = (n + ):ψ nψm+nηab :+ (r + ):ψ rψm+r : (31) 2 2 − 2 2 − n Z r Z+ 1 X∈ ∈X 2 F,R 1 m a b 1 m i i Lm = (r + ):ψ rψm+rηab :+ (n + ):ψ nψm+n : (32) 2 2 − 2 2 − r Z+ 1 n Z ∈X 2 X∈ The super current components in the twisted sector are

a b i i Gr = : ψ mαr+mηab + ψr+mα m : (33) m − − X a b i i Fn = : ψ m+ 1 αn+m 1 ηab + ψn+mα m : . (34) m − 2 − 2 − X There are similar expressions for the left-movers.

18 B The twisted sector vacua as spinors

Consider 2n worldsheet fermion zero modes ψa satisfying ψa,ψb = δab and commut- 0 { 0 0} ing with the mass operator. The theory then has an SO(2n) spinor as its degenerate a a vacuum, which can be constructed as follows. Deﬁne Γ = √2ψ0 . These then satisfy the Dirac algebra Γa, Γb =2δab. The ground state is a representation of this algebra { } and can be constructed using the standard procedure: For k =1, n, deﬁne ··· 1 1 ek = (Γk + iΓn k) ,e† = (Γk iΓn k) . 2 + k 2 − +

These satisfy the fermionic algebra, e† ,el = δkl, with other anti-commutators { k } vanishing. Start from a state 0 annihilated by the lowering operators. Other com- | i ponents of the ground state spinor are obtained by using the raising operators on this lowest state; 0 ,ek† 0 ,ek† el† 0 , ,e1†e2† n† 0 . The degeneracy of a state with p | i | i | i ··· ···n | i n raising operators is the combinatoric factor, Cp and the total number of states is 2 ; that of a spinor. The chirality operator is Γ n =Γ Γ n. States with even (odd) 2 +1 1 ··· 2 number of oscillators form a positive (negative) chirality spinor. In the twisted NS sector, 2n = d + 1 and in the twisted R sector 2n =9 d. − References

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